Question:
Evaluate the following integrals:
$\int \cos \sqrt{x} d x$
Solution:
Let $I=\int \cos \sqrt{x} d x$
$\sqrt{\mathrm{X}}=\mathrm{t} ; \mathrm{x}=\mathrm{t}^{2}$
$d x=2 t d t$
$=\int 2 t \cos t d t$
$I=2 \int t \cos t d t$
Using integration by parts,
$=2\left(\mathrm{t} \times \sin \mathrm{t}-\int \sin \mathrm{t} \mathrm{dt}\right)$
$=2(\mathrm{t} \sin \mathrm{t}+\cos \mathrm{t})+\mathrm{c}$
Replace the value of $t, I=2(\sqrt{x} \sin \sqrt{x}+\cos \sqrt{x})+c$
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